+ 
202 
POWER SERIES 
Thus F give rise to the double series 
I) = ^4-qq + A^(x Xq) + A.q%(x ^q) 2 + 
+ A 'l0 + ^ilO - x o) + A 'n(. x - x o) 2 + 
+ 
where 
- (-!)” a 
. te • 
n 1 
The adjoint series to f n (x) is, setting £= \ x — a? 0 1 
<£»(£) = 
^ + 
a 2»| 2 
,„ ! \1 + a n :z 0 (1 -f a n x ü ) 2 (1 + a n x 0 ) 3 
This is convergent if 
<1 or if £<x 0 , 
that is, if 
1 + a n x 0 
0<x<2x 0 , 
For any x such that x 0 < x < 2 x Q , f; = x — x 0 . 
Then for such an x 
</>» = * 
n ! 1 + a n (2 x Q — x) 
and the corresponding series 
=£(/>„ 
is evidently convergent, since $„< — ■ 
n ! 
We may thus sum D by columns; we get 
F(x) = 2 B K (x - x 0 ) 
where 
K=0 
_ y(-l) n+K a nK 
»=0 "■ ! n + «%,)'*' 
The relation 1) is valid for 0 < x < 2 æ 0 .